> So one obvious thought is, I could, perhaps for certain problems, teach my student to think like me. And I did a little of that. She wondered why multiplying by a number less than one would make something smaller (because MULTIPLYING makes things BIGGER, right?). I created an animation.. three bars going up and down. The left two bars are the multiplicands, and the right bar is the result. Usually the middle bar is fixed, and the leftmost one varies between 0 and 2, passing through 1 on the way up and again on the way down. My student could see that as the left bar approached 1, the result approached the fixed center bar -- and she already knew that "anything multiplied by 1 is itself," so this confirmed it. When the bar dipped below 1, it made complete sense that the result bar would go down and get smaller than the fixed multiplicand. And when the left bar got to 0, then you could see WHY "anything multiplied by 0 is 0."

If you have actually done this, succeeded at teaching this student multiplication through visuals then you need to provide proof that she indeed understands multiplication. You need to have her work out an array of problems involving multiplication, division and fractions. I have not found any evidence at all that any student gets mathematics "visually". They may get mathematics and there may be visuals involved, but no person I have interviewed has ever shown me evidence of applicable mathematical sense based on vision. That is like telling me you believe in psychics.

> Right now I'm working on making a series of videos which could turn into computer software, teaching how to recognize the "form" of an algebraic expression or equation, and teaching why it's useful and powerful to be able to transform an equation.

That is like saying that you have taught someone that keeping your eye on the ball is important. That isn't saying that they actually keep their eye on the ball, just that they know it is important. But how can anyone say they "know" that it is important without actually doing it? There is only one way to succeed at teaching someone that the form of an algebraic expression is useful, and that is while THEY are doing algebra they recognize what form and convention brings to the task. These are not things you can take out of context and say you have taught it.

> But what of thinkers who don't feel drawn to this style? Let's say a student comes to me who has a hard time in math. (Like my current student.) And they reveal some of the their learning style and it is not an adaptive style for math (like my current student). Is it my job, then, to help them do math from their current perspective, or is it my job to introduce them to a much more powerful perspective, even if they only get a little bit of it, long enough to pass math class?

I have tutored students and I know exactly what you are going through. I try to teach them "math" but I admit that I know soon enough whether we will strive for math or strive for passing the class. And I adjust my style depending on which goal it is. And if it is the later and the pictures are enough to pass the class then so be it. Personally, I try to focus on a small subset of rules that fit together and that they generally have a high probability of remembering, at least long enough to take the test. Generally these are not students taking advanced classes so that works well, although this was many years ago and today these same students are actually taking advanced classes. And I would give pointers like that even if I wasn't being paid. We all went to school. We all know what it feels like just to want to pass the class. It's just a nice thing to do.